Abstract

Among the circle of conjectures which forms the global Langlands correspondence, perhaps the simplest is the prediction that the Hecke eigenvalues of L and C-algebraic automorphic representations \pi are algebraic numbers. Fundamental to our current understanding of this conjecture is a dictionary between representation-theoretic properties of the archimedean component \pi_{\infty}, e.g. “non-degenerate or degenerate limit of discrete series” (LDS), and geometric properties of \pi, e.g. “appears in the coherent cohomology of a Shimura variety or Griffiths-Schmid manifold”.

We propose a systematic study of the conjectural implications of Langlands functoriality to the above conjecture. To this end, we study the (in)variance of the dichotomies “LDS/non-LDS” and “non-degenerate/degenerate” under functoriality. In the positive direction, we give examples where functoriality implies new cases of algebraicity and (work in progress) show that one class of these follows unconditionally from Arthur’s work on endoscopy.